Optimal. Leaf size=77 \[ \frac{-a B e-A b e+2 b B d}{3 e^3 (d+e x)^3}-\frac{(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}-\frac{b B}{2 e^3 (d+e x)^2} \]
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Rubi [A] time = 0.0494804, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ \frac{-a B e-A b e+2 b B d}{3 e^3 (d+e x)^3}-\frac{(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}-\frac{b B}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x) (A+B x)}{(d+e x)^5} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e)}{e^2 (d+e x)^5}+\frac{-2 b B d+A b e+a B e}{e^2 (d+e x)^4}+\frac{b B}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac{(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}+\frac{2 b B d-A b e-a B e}{3 e^3 (d+e x)^3}-\frac{b B}{2 e^3 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0303167, size = 62, normalized size = 0.81 \[ -\frac{a e (3 A e+B (d+4 e x))+b \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )}{12 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 79, normalized size = 1. \begin{align*} -{\frac{Abe+Bae-2\,Bbd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Bb}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{aA{e}^{2}-Adbe-Bdae+bB{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04274, size = 138, normalized size = 1.79 \begin{align*} -\frac{6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} +{\left (B a + A b\right )} d e + 4 \,{\left (B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77576, size = 217, normalized size = 2.82 \begin{align*} -\frac{6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} +{\left (B a + A b\right )} d e + 4 \,{\left (B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.45697, size = 117, normalized size = 1.52 \begin{align*} - \frac{3 A a e^{2} + A b d e + B a d e + B b d^{2} + 6 B b e^{2} x^{2} + x \left (4 A b e^{2} + 4 B a e^{2} + 4 B b d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.97165, size = 163, normalized size = 2.12 \begin{align*} -\frac{1}{12} \,{\left (\frac{6 \, B b e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{8 \, B b d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac{3 \, B b d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} + \frac{4 \, A b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} - \frac{3 \, A b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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